nprmdvdsfacm1lem4 - Mathbox for Alexander van der Vekens

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Theorem nprmdvdsfacm1lem4 48080

Description: Lemma 4 for nprmdvdsfacm1 48081. (Contributed by AV, 7-Apr-2026.)

**Assertion**
Ref	Expression
nprmdvdsfacm1lem4	⊢ ((𝑁 ∈ (ℤ_≥‘6) ∧ 𝐴 ∈ (2..^𝑁) ∧ 𝑁 = (𝐴↑2)) → 𝑁 ∥ (!‘(𝑁 − 1)))

**Proof of Theorem nprmdvdsfacm1lem4**
Step	Hyp	Ref	Expression
1		eluzelz 12798	. . 3 ⊢ (𝑁 ∈ (ℤ_≥‘6) → 𝑁 ∈ ℤ)
2	1	3ad2ant1 1134	. 2 ⊢ ((𝑁 ∈ (ℤ_≥‘6) ∧ 𝐴 ∈ (2..^𝑁) ∧ 𝑁 = (𝐴↑2)) → 𝑁 ∈ ℤ)
3		elfzoelz 13613	. . . 4 ⊢ (𝐴 ∈ (2..^𝑁) → 𝐴 ∈ ℤ)
4		id 22	. . . . 5 ⊢ (𝐴 ∈ ℤ → 𝐴 ∈ ℤ)
5		2z 12559	. . . . . . 7 ⊢ 2 ∈ ℤ
6	5	a1i 11	. . . . . 6 ⊢ (𝐴 ∈ ℤ → 2 ∈ ℤ)
7	6, 4	zmulcld 12639	. . . . 5 ⊢ (𝐴 ∈ ℤ → (2 · 𝐴) ∈ ℤ)
8	4, 7	zmulcld 12639	. . . 4 ⊢ (𝐴 ∈ ℤ → (𝐴 · (2 · 𝐴)) ∈ ℤ)
9	3, 8	syl 17	. . 3 ⊢ (𝐴 ∈ (2..^𝑁) → (𝐴 · (2 · 𝐴)) ∈ ℤ)
10	9	3ad2ant2 1135	. 2 ⊢ ((𝑁 ∈ (ℤ_≥‘6) ∧ 𝐴 ∈ (2..^𝑁) ∧ 𝑁 = (𝐴↑2)) → (𝐴 · (2 · 𝐴)) ∈ ℤ)
11		1z 12557	. . . . . . . . . 10 ⊢ 1 ∈ ℤ
12		6nn 12270	. . . . . . . . . . 11 ⊢ 6 ∈ ℕ
13	12	nnzi 12551	. . . . . . . . . 10 ⊢ 6 ∈ ℤ
14		1re 11144	. . . . . . . . . . 11 ⊢ 1 ∈ ℝ
15		6re 12271	. . . . . . . . . . 11 ⊢ 6 ∈ ℝ
16		1lt6 12361	. . . . . . . . . . 11 ⊢ 1 < 6
17	14, 15, 16	ltleii 11269	. . . . . . . . . 10 ⊢ 1 ≤ 6
18		eluz2 12794	. . . . . . . . . 10 ⊢ (6 ∈ (ℤ_≥‘1) ↔ (1 ∈ ℤ ∧ 6 ∈ ℤ ∧ 1 ≤ 6))
19	11, 13, 17, 18	mpbir3an 1343	. . . . . . . . 9 ⊢ 6 ∈ (ℤ_≥‘1)
20		uzss 12811	. . . . . . . . 9 ⊢ (6 ∈ (ℤ_≥‘1) → (ℤ_≥‘6) ⊆ (ℤ_≥‘1))
21	19, 20	ax-mp 5	. . . . . . . 8 ⊢ (ℤ_≥‘6) ⊆ (ℤ_≥‘1)
22	21	sseli 3918	. . . . . . 7 ⊢ (𝑁 ∈ (ℤ_≥‘6) → 𝑁 ∈ (ℤ_≥‘1))
23		elnnuz 12828	. . . . . . 7 ⊢ (𝑁 ∈ ℕ ↔ 𝑁 ∈ (ℤ_≥‘1))
24	22, 23	sylibr 234	. . . . . 6 ⊢ (𝑁 ∈ (ℤ_≥‘6) → 𝑁 ∈ ℕ)
25	24	3ad2ant1 1134	. . . . 5 ⊢ ((𝑁 ∈ (ℤ_≥‘6) ∧ 𝐴 ∈ (2..^𝑁) ∧ 𝑁 = (𝐴↑2)) → 𝑁 ∈ ℕ)
26		nnm1nn0 12478	. . . . 5 ⊢ (𝑁 ∈ ℕ → (𝑁 − 1) ∈ ℕ₀)
27	25, 26	syl 17	. . . 4 ⊢ ((𝑁 ∈ (ℤ_≥‘6) ∧ 𝐴 ∈ (2..^𝑁) ∧ 𝑁 = (𝐴↑2)) → (𝑁 − 1) ∈ ℕ₀)
28	27	faccld 14246	. . 3 ⊢ ((𝑁 ∈ (ℤ_≥‘6) ∧ 𝐴 ∈ (2..^𝑁) ∧ 𝑁 = (𝐴↑2)) → (!‘(𝑁 − 1)) ∈ ℕ)
29	28	nnzd 12550	. 2 ⊢ ((𝑁 ∈ (ℤ_≥‘6) ∧ 𝐴 ∈ (2..^𝑁) ∧ 𝑁 = (𝐴↑2)) → (!‘(𝑁 − 1)) ∈ ℤ)
30		nprmdvdsfacm1lem1 48077	. 2 ⊢ ((𝑁 ∈ (ℤ_≥‘6) ∧ 𝐴 ∈ (2..^𝑁) ∧ 𝑁 = (𝐴↑2)) → 𝑁 ∥ (𝐴 · (2 · 𝐴)))
31		elfzo2 13616	. . . . . 6 ⊢ (𝐴 ∈ (2..^𝑁) ↔ (𝐴 ∈ (ℤ_≥‘2) ∧ 𝑁 ∈ ℤ ∧ 𝐴 < 𝑁))
32		2eluzge1 12832	. . . . . . . . . 10 ⊢ 2 ∈ (ℤ_≥‘1)
33		uzss 12811	. . . . . . . . . 10 ⊢ (2 ∈ (ℤ_≥‘1) → (ℤ_≥‘2) ⊆ (ℤ_≥‘1))
34	32, 33	ax-mp 5	. . . . . . . . 9 ⊢ (ℤ_≥‘2) ⊆ (ℤ_≥‘1)
35	34	sseli 3918	. . . . . . . 8 ⊢ (𝐴 ∈ (ℤ_≥‘2) → 𝐴 ∈ (ℤ_≥‘1))
36	35	3ad2ant1 1134	. . . . . . 7 ⊢ ((𝐴 ∈ (ℤ_≥‘2) ∧ 𝑁 ∈ ℤ ∧ 𝐴 < 𝑁) → 𝐴 ∈ (ℤ_≥‘1))
37	5	a1i 11	. . . . . . . . 9 ⊢ (𝐴 ∈ (ℤ_≥‘2) → 2 ∈ ℤ)
38		eluzelz 12798	. . . . . . . . 9 ⊢ (𝐴 ∈ (ℤ_≥‘2) → 𝐴 ∈ ℤ)
39	37, 38	zmulcld 12639	. . . . . . . 8 ⊢ (𝐴 ∈ (ℤ_≥‘2) → (2 · 𝐴) ∈ ℤ)
40	39	3ad2ant1 1134	. . . . . . 7 ⊢ ((𝐴 ∈ (ℤ_≥‘2) ∧ 𝑁 ∈ ℤ ∧ 𝐴 < 𝑁) → (2 · 𝐴) ∈ ℤ)
41		1lt2 12347	. . . . . . . 8 ⊢ 1 < 2
42		eluz2 12794	. . . . . . . . . . 11 ⊢ (𝐴 ∈ (ℤ_≥‘2) ↔ (2 ∈ ℤ ∧ 𝐴 ∈ ℤ ∧ 2 ≤ 𝐴))
43		zre 12528	. . . . . . . . . . . . 13 ⊢ (𝐴 ∈ ℤ → 𝐴 ∈ ℝ)
44	43	3ad2ant2 1135	. . . . . . . . . . . 12 ⊢ ((2 ∈ ℤ ∧ 𝐴 ∈ ℤ ∧ 2 ≤ 𝐴) → 𝐴 ∈ ℝ)
45		2re 12255	. . . . . . . . . . . . 13 ⊢ 2 ∈ ℝ
46	45	a1i 11	. . . . . . . . . . . 12 ⊢ ((2 ∈ ℤ ∧ 𝐴 ∈ ℤ ∧ 2 ≤ 𝐴) → 2 ∈ ℝ)
47		2pos 12284	. . . . . . . . . . . . . . 15 ⊢ 0 < 2
48		0red 11147	. . . . . . . . . . . . . . . . 17 ⊢ (𝐴 ∈ ℤ → 0 ∈ ℝ)
49	45	a1i 11	. . . . . . . . . . . . . . . . 17 ⊢ (𝐴 ∈ ℤ → 2 ∈ ℝ)
50	48, 49, 43	3jca 1129	. . . . . . . . . . . . . . . 16 ⊢ (𝐴 ∈ ℤ → (0 ∈ ℝ ∧ 2 ∈ ℝ ∧ 𝐴 ∈ ℝ))
51		ltletr 11238	. . . . . . . . . . . . . . . 16 ⊢ ((0 ∈ ℝ ∧ 2 ∈ ℝ ∧ 𝐴 ∈ ℝ) → ((0 < 2 ∧ 2 ≤ 𝐴) → 0 < 𝐴))
52	50, 51	syl 17	. . . . . . . . . . . . . . 15 ⊢ (𝐴 ∈ ℤ → ((0 < 2 ∧ 2 ≤ 𝐴) → 0 < 𝐴))
53	47, 52	mpani 697	. . . . . . . . . . . . . 14 ⊢ (𝐴 ∈ ℤ → (2 ≤ 𝐴 → 0 < 𝐴))
54	53	imp 406	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ ℤ ∧ 2 ≤ 𝐴) → 0 < 𝐴)
55	54	3adant1 1131	. . . . . . . . . . . 12 ⊢ ((2 ∈ ℤ ∧ 𝐴 ∈ ℤ ∧ 2 ≤ 𝐴) → 0 < 𝐴)
56	44, 46, 55	3jca 1129	. . . . . . . . . . 11 ⊢ ((2 ∈ ℤ ∧ 𝐴 ∈ ℤ ∧ 2 ≤ 𝐴) → (𝐴 ∈ ℝ ∧ 2 ∈ ℝ ∧ 0 < 𝐴))
57	42, 56	sylbi 217	. . . . . . . . . 10 ⊢ (𝐴 ∈ (ℤ_≥‘2) → (𝐴 ∈ ℝ ∧ 2 ∈ ℝ ∧ 0 < 𝐴))
58	57	3ad2ant1 1134	. . . . . . . . 9 ⊢ ((𝐴 ∈ (ℤ_≥‘2) ∧ 𝑁 ∈ ℤ ∧ 𝐴 < 𝑁) → (𝐴 ∈ ℝ ∧ 2 ∈ ℝ ∧ 0 < 𝐴))
59		ltmulgt12 12016	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℝ ∧ 2 ∈ ℝ ∧ 0 < 𝐴) → (1 < 2 ↔ 𝐴 < (2 · 𝐴)))
60	58, 59	syl 17	. . . . . . . 8 ⊢ ((𝐴 ∈ (ℤ_≥‘2) ∧ 𝑁 ∈ ℤ ∧ 𝐴 < 𝑁) → (1 < 2 ↔ 𝐴 < (2 · 𝐴)))
61	41, 60	mpbii 233	. . . . . . 7 ⊢ ((𝐴 ∈ (ℤ_≥‘2) ∧ 𝑁 ∈ ℤ ∧ 𝐴 < 𝑁) → 𝐴 < (2 · 𝐴))
62	36, 40, 61	3jca 1129	. . . . . 6 ⊢ ((𝐴 ∈ (ℤ_≥‘2) ∧ 𝑁 ∈ ℤ ∧ 𝐴 < 𝑁) → (𝐴 ∈ (ℤ_≥‘1) ∧ (2 · 𝐴) ∈ ℤ ∧ 𝐴 < (2 · 𝐴)))
63	31, 62	sylbi 217	. . . . 5 ⊢ (𝐴 ∈ (2..^𝑁) → (𝐴 ∈ (ℤ_≥‘1) ∧ (2 · 𝐴) ∈ ℤ ∧ 𝐴 < (2 · 𝐴)))
64		elfzo2 13616	. . . . 5 ⊢ (𝐴 ∈ (1..^(2 · 𝐴)) ↔ (𝐴 ∈ (ℤ_≥‘1) ∧ (2 · 𝐴) ∈ ℤ ∧ 𝐴 < (2 · 𝐴)))
65	63, 64	sylibr 234	. . . 4 ⊢ (𝐴 ∈ (2..^𝑁) → 𝐴 ∈ (1..^(2 · 𝐴)))
66	65	3ad2ant2 1135	. . 3 ⊢ ((𝑁 ∈ (ℤ_≥‘6) ∧ 𝐴 ∈ (2..^𝑁) ∧ 𝑁 = (𝐴↑2)) → 𝐴 ∈ (1..^(2 · 𝐴)))
67	11	a1i 11	. . . . . 6 ⊢ ((𝑁 ∈ (ℤ_≥‘6) ∧ 𝐴 ∈ (2..^𝑁) ∧ 𝑁 = (𝐴↑2)) → 1 ∈ ℤ)
68	5	a1i 11	. . . . . . . 8 ⊢ (𝐴 ∈ (2..^𝑁) → 2 ∈ ℤ)
69	68, 3	zmulcld 12639	. . . . . . 7 ⊢ (𝐴 ∈ (2..^𝑁) → (2 · 𝐴) ∈ ℤ)
70	69	3ad2ant2 1135	. . . . . 6 ⊢ ((𝑁 ∈ (ℤ_≥‘6) ∧ 𝐴 ∈ (2..^𝑁) ∧ 𝑁 = (𝐴↑2)) → (2 · 𝐴) ∈ ℤ)
71	47	a1i 11	. . . . . . . . . . . . 13 ⊢ (𝐴 ∈ ℤ → 0 < 2)
72		lemul2 12008	. . . . . . . . . . . . 13 ⊢ ((2 ∈ ℝ ∧ 𝐴 ∈ ℝ ∧ (2 ∈ ℝ ∧ 0 < 2)) → (2 ≤ 𝐴 ↔ (2 · 2) ≤ (2 · 𝐴)))
73	49, 43, 49, 71, 72	syl112anc 1377	. . . . . . . . . . . 12 ⊢ (𝐴 ∈ ℤ → (2 ≤ 𝐴 ↔ (2 · 2) ≤ (2 · 𝐴)))
74		1red 11145	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ ℤ ∧ (2 · 2) ≤ (2 · 𝐴)) → 1 ∈ ℝ)
75		2t2e4 12340	. . . . . . . . . . . . . . 15 ⊢ (2 · 2) = 4
76		4re 12265	. . . . . . . . . . . . . . 15 ⊢ 4 ∈ ℝ
77	75, 76	eqeltri 2833	. . . . . . . . . . . . . 14 ⊢ (2 · 2) ∈ ℝ
78	77	a1i 11	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ ℤ ∧ (2 · 2) ≤ (2 · 𝐴)) → (2 · 2) ∈ ℝ)
79	7	zred 12633	. . . . . . . . . . . . . 14 ⊢ (𝐴 ∈ ℤ → (2 · 𝐴) ∈ ℝ)
80	79	adantr 480	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ ℤ ∧ (2 · 2) ≤ (2 · 𝐴)) → (2 · 𝐴) ∈ ℝ)
81		1lt4 12352	. . . . . . . . . . . . . . . 16 ⊢ 1 < 4
82	14, 76, 81	ltleii 11269	. . . . . . . . . . . . . . 15 ⊢ 1 ≤ 4
83	82, 75	breqtrri 5113	. . . . . . . . . . . . . 14 ⊢ 1 ≤ (2 · 2)
84	83	a1i 11	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ ℤ ∧ (2 · 2) ≤ (2 · 𝐴)) → 1 ≤ (2 · 2))
85		simpr 484	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ ℤ ∧ (2 · 2) ≤ (2 · 𝐴)) → (2 · 2) ≤ (2 · 𝐴))
86	74, 78, 80, 84, 85	letrd 11303	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ ℤ ∧ (2 · 2) ≤ (2 · 𝐴)) → 1 ≤ (2 · 𝐴))
87	73, 86	sylbida 593	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ ℤ ∧ 2 ≤ 𝐴) → 1 ≤ (2 · 𝐴))
88	87	3adant1 1131	. . . . . . . . . 10 ⊢ ((2 ∈ ℤ ∧ 𝐴 ∈ ℤ ∧ 2 ≤ 𝐴) → 1 ≤ (2 · 𝐴))
89	42, 88	sylbi 217	. . . . . . . . 9 ⊢ (𝐴 ∈ (ℤ_≥‘2) → 1 ≤ (2 · 𝐴))
90	89	3ad2ant1 1134	. . . . . . . 8 ⊢ ((𝐴 ∈ (ℤ_≥‘2) ∧ 𝑁 ∈ ℤ ∧ 𝐴 < 𝑁) → 1 ≤ (2 · 𝐴))
91	31, 90	sylbi 217	. . . . . . 7 ⊢ (𝐴 ∈ (2..^𝑁) → 1 ≤ (2 · 𝐴))
92	91	3ad2ant2 1135	. . . . . 6 ⊢ ((𝑁 ∈ (ℤ_≥‘6) ∧ 𝐴 ∈ (2..^𝑁) ∧ 𝑁 = (𝐴↑2)) → 1 ≤ (2 · 𝐴))
93		eluz2 12794	. . . . . 6 ⊢ ((2 · 𝐴) ∈ (ℤ_≥‘1) ↔ (1 ∈ ℤ ∧ (2 · 𝐴) ∈ ℤ ∧ 1 ≤ (2 · 𝐴)))
94	67, 70, 92, 93	syl3anbrc 1345	. . . . 5 ⊢ ((𝑁 ∈ (ℤ_≥‘6) ∧ 𝐴 ∈ (2..^𝑁) ∧ 𝑁 = (𝐴↑2)) → (2 · 𝐴) ∈ (ℤ_≥‘1))
95		zsqcl 14091	. . . . . . 7 ⊢ (𝐴 ∈ ℤ → (𝐴↑2) ∈ ℤ)
96	3, 95	syl 17	. . . . . 6 ⊢ (𝐴 ∈ (2..^𝑁) → (𝐴↑2) ∈ ℤ)
97	96	3ad2ant2 1135	. . . . 5 ⊢ ((𝑁 ∈ (ℤ_≥‘6) ∧ 𝐴 ∈ (2..^𝑁) ∧ 𝑁 = (𝐴↑2)) → (𝐴↑2) ∈ ℤ)
98		3z 12560	. . . . . . . 8 ⊢ 3 ∈ ℤ
99	98	a1i 11	. . . . . . 7 ⊢ ((𝑁 ∈ (ℤ_≥‘6) ∧ 𝐴 ∈ (2..^𝑁) ∧ 𝑁 = (𝐴↑2)) → 3 ∈ ℤ)
100	3	3ad2ant2 1135	. . . . . . 7 ⊢ ((𝑁 ∈ (ℤ_≥‘6) ∧ 𝐴 ∈ (2..^𝑁) ∧ 𝑁 = (𝐴↑2)) → 𝐴 ∈ ℤ)
101		nprmdvdsfacm1lem2 48078	. . . . . . 7 ⊢ ((𝑁 ∈ (ℤ_≥‘6) ∧ 𝐴 ∈ (2..^𝑁) ∧ 𝑁 = (𝐴↑2)) → 3 ≤ 𝐴)
102		eluz2 12794	. . . . . . 7 ⊢ (𝐴 ∈ (ℤ_≥‘3) ↔ (3 ∈ ℤ ∧ 𝐴 ∈ ℤ ∧ 3 ≤ 𝐴))
103	99, 100, 101, 102	syl3anbrc 1345	. . . . . 6 ⊢ ((𝑁 ∈ (ℤ_≥‘6) ∧ 𝐴 ∈ (2..^𝑁) ∧ 𝑁 = (𝐴↑2)) → 𝐴 ∈ (ℤ_≥‘3))
104		2timesltsq 47820	. . . . . 6 ⊢ (𝐴 ∈ (ℤ_≥‘3) → (2 · 𝐴) < (𝐴↑2))
105	103, 104	syl 17	. . . . 5 ⊢ ((𝑁 ∈ (ℤ_≥‘6) ∧ 𝐴 ∈ (2..^𝑁) ∧ 𝑁 = (𝐴↑2)) → (2 · 𝐴) < (𝐴↑2))
106	94, 97, 105	elfzod 13617	. . . 4 ⊢ ((𝑁 ∈ (ℤ_≥‘6) ∧ 𝐴 ∈ (2..^𝑁) ∧ 𝑁 = (𝐴↑2)) → (2 · 𝐴) ∈ (1..^(𝐴↑2)))
107		oveq2 7375	. . . . . 6 ⊢ (𝑁 = (𝐴↑2) → (1..^𝑁) = (1..^(𝐴↑2)))
108	107	eleq2d 2823	. . . . 5 ⊢ (𝑁 = (𝐴↑2) → ((2 · 𝐴) ∈ (1..^𝑁) ↔ (2 · 𝐴) ∈ (1..^(𝐴↑2))))
109	108	3ad2ant3 1136	. . . 4 ⊢ ((𝑁 ∈ (ℤ_≥‘6) ∧ 𝐴 ∈ (2..^𝑁) ∧ 𝑁 = (𝐴↑2)) → ((2 · 𝐴) ∈ (1..^𝑁) ↔ (2 · 𝐴) ∈ (1..^(𝐴↑2))))
110	106, 109	mpbird 257	. . 3 ⊢ ((𝑁 ∈ (ℤ_≥‘6) ∧ 𝐴 ∈ (2..^𝑁) ∧ 𝑁 = (𝐴↑2)) → (2 · 𝐴) ∈ (1..^𝑁))
111		muldvdsfacm1 47829	. . 3 ⊢ ((𝐴 ∈ (1..^(2 · 𝐴)) ∧ (2 · 𝐴) ∈ (1..^𝑁)) → (𝐴 · (2 · 𝐴)) ∥ (!‘(𝑁 − 1)))
112	66, 110, 111	syl2anc 585	. 2 ⊢ ((𝑁 ∈ (ℤ_≥‘6) ∧ 𝐴 ∈ (2..^𝑁) ∧ 𝑁 = (𝐴↑2)) → (𝐴 · (2 · 𝐴)) ∥ (!‘(𝑁 − 1)))
113	2, 10, 29, 30, 112	dvdstrd 16264	1 ⊢ ((𝑁 ∈ (ℤ_≥‘6) ∧ 𝐴 ∈ (2..^𝑁) ∧ 𝑁 = (𝐴↑2)) → 𝑁 ∥ (!‘(𝑁 − 1)))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 ∧ w3a 1087 = wceq 1542 ∈ wcel 2114 ⊆ wss 3890 class class class wbr 5086 ‘cfv 6499 (class class class)co 7367 ℝcr 11037 0cc0 11038 1c1 11039 · cmul 11043 < clt 11179 ≤ cle 11180 − cmin 11377 ℕcn 12174 2c2 12236 3c3 12237 4c4 12238 6c6 12240 ℕ₀cn0 12437 ℤcz 12524 ℤ_≥cuz 12788 ..^cfzo 13608 ↑cexp 14023 !cfa 14235 ∥ cdvds 16221

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1912 ax-6 1969 ax-7 2010 ax-8 2116 ax-9 2124 ax-10 2147 ax-11 2163 ax-12 2185 ax-ext 2709 ax-sep 5232 ax-nul 5242 ax-pow 5308 ax-pr 5376 ax-un 7689 ax-cnex 11094 ax-resscn 11095 ax-1cn 11096 ax-icn 11097 ax-addcl 11098 ax-addrcl 11099 ax-mulcl 11100 ax-mulrcl 11101 ax-mulcom 11102 ax-addass 11103 ax-mulass 11104 ax-distr 11105 ax-i2m1 11106 ax-1ne0 11107 ax-1rid 11108 ax-rnegex 11109 ax-rrecex 11110 ax-cnre 11111 ax-pre-lttri 11112 ax-pre-lttrn 11113 ax-pre-ltadd 11114 ax-pre-mulgt0 11115

This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1545 df-fal 1555 df-ex 1782 df-nf 1786 df-sb 2069 df-mo 2540 df-eu 2570 df-clab 2716 df-cleq 2729 df-clel 2812 df-nfc 2886 df-ne 2934 df-nel 3038 df-ral 3053 df-rex 3063 df-rmo 3343 df-reu 3344 df-rab 3391 df-v 3432 df-sbc 3730 df-csb 3839 df-dif 3893 df-un 3895 df-in 3897 df-ss 3907 df-pss 3910 df-nul 4275 df-if 4468 df-pw 4544 df-sn 4569 df-pr 4571 df-op 4575 df-uni 4852 df-iun 4936 df-br 5087 df-opab 5149 df-mpt 5168 df-tr 5194 df-id 5526 df-eprel 5531 df-po 5539 df-so 5540 df-fr 5584 df-we 5586 df-xp 5637 df-rel 5638 df-cnv 5639 df-co 5640 df-dm 5641 df-rn 5642 df-res 5643 df-ima 5644 df-pred 6266 df-ord 6327 df-on 6328 df-lim 6329 df-suc 6330 df-iota 6455 df-fun 6501 df-fn 6502 df-f 6503 df-f1 6504 df-fo 6505 df-f1o 6506 df-fv 6507 df-riota 7324 df-ov 7370 df-oprab 7371 df-mpo 7372 df-om 7818 df-1st 7942 df-2nd 7943 df-frecs 8231 df-wrecs 8262 df-recs 8311 df-rdg 8349 df-er 8643 df-en 8894 df-dom 8895 df-sdom 8896 df-pnf 11181 df-mnf 11182 df-xr 11183 df-ltxr 11184 df-le 11185 df-sub 11379 df-neg 11380 df-div 11808 df-nn 12175 df-2 12244 df-3 12245 df-4 12246 df-5 12247 df-6 12248 df-n0 12438 df-z 12525 df-uz 12789 df-rp 12943 df-fz 13462 df-fzo 13609 df-seq 13964 df-exp 14024 df-fac 14236 df-bc 14265 df-dvds 16222

This theorem is referenced by: nprmdvdsfacm1 48081

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